We consider the coupled chemotaxis-fluid model for periodic pattern formation on two- and three-dimensional domains with mixed nonhomogeneous boundary value conditions, and prove the existence of nontrivial time periodic solutions. It is worth noticing that this system admits more than one periodic solution. In fact, it is not difficult to verify that (0, c, 0, 0) is a time periodic solution. Our purpose is to obtain a time periodic solution with nonconstant bacterial density.

中文翻译：

---

具有混合非均匀边界条件的耦合趋化性-(Navier-)Stokes 系统中的周期性模式形成

我们考虑了具有混合非齐次边值条件的二维和三维域上周期性模式形成的耦合趋化流体模型，并证明了非平凡时间周期解的存在。值得注意的是，该系统允许多个周期解。其实不难验证 (0,C, 0, 0) 是时间周期解。我们的目的是获得具有非恒定细菌密度的时间周期解。